The n-Category Café

Yes, certainly. I’m sure one of us nLab authors will get to it by and by.

I have created an entry on Kan lifts at the nLab.

Sorry, it’s not quite right. But if $f$ preserves the lift, then I think it’s right (I’ll explain in a minute).

As I was saying below, right Kan lifts (Rifts) of bimodules between small categories (and again let’s work with Cauchy complete ones) always exist. But that doesn’t mean ${\mathrm{Rift}}_f{1}_B$ will be the right adjoint of $f:A\to B$, because $f:A\to B$ may not be a left adjoint. (It is so when the bimodule is given by a ring map $f:A\to B$.)

The best you get out of the bargain is that you get a canonical arrow

$$\epsilon :f\circ {\mathrm{Rift}}_f{1}_B\Rightarrow 1_B$$

which would be the counit, if $f⊣{\mathrm{Rift}}_f{1}_B$.

However, there is a canonical map

$$\theta :({\mathrm{Rift}}_f{1}_B)\circ f\Rightarrow {\mathrm{Rift}}_{f}f$$

corresponding to the map

$$\epsilon \circ f:(f\circ {\mathrm{Rift}}_f{1}_B)\circ f\Rightarrow 1_B\circ f=f$$

and it is also true that ${\mathrm{Rift}}_{f}f$ is a monad. If $\theta$ above is an isomorphism (in which case we say that $f$ preserves the right Kan lift), then I think we have an adjunction $f⊣{\mathrm{Rift}}_f{1}_B$.

(I’m going on memory here without checking it carefully; maybe someone should check up on that.)

I said earlier

But that doesn’t mean ${\mathrm{Rift}}_f{1}_B$ will be the right adjoint of $f:A\to B$, because $f:A\to B$ may not be a left adjoint. (It is so when the bimodule is given by a ring map $f:A\to B$.)

I should have said, “when the bimodule is given by a functor $f:A\to B$”.

My point was that in the bicategory of small categories and bimodules between them, left adjoints $r:A\to B$ in the bicategory are essentially the same as bimodules of the form

$${\mathrm{hom}}_B(-,f-):B^{\mathrm{op}}\times A\to \mathrm{Set}$$

where $f:A\to B$ is a functor. Technically, that’s not quite correct, unless the categories in question are Cauchy complete, which is why I threw in that assumption.

SH: I’ve mentioned before that defining causality can be seen as the core issue:

http://en.wikipedia.org/wiki/Causality
“The Nobel Prize holder Herbert Simon and Philosopher Nicholas Rescher[20]
claim that the asymmetry of the causal relation is unrelated to the asymmetry
of any mode of implication that contraposes. Rather, a causal relation is not
a relation between values of variables, but a function of one variable (the
cause) on to another (the effect). So, given a system of equations, and a set
of variables appearing in these equations, we can introduce an asymmetric
relation among individual equations and variables that corresponds perfectly
to our commonsense notion of a causal ordering. The system of equations must
have certain properties, most importantly, if some values are chosen arbitrarily,
the remaining values will be determined uniquely through a path of serial
discovery that is perfectly causal. They postulate the inherent serialization of
such a system of equations may correctly capture causation in all empirical fields,
including physics and economics.”
——————————————

Todd Trimble wrote Re: Kan Lifts
“Wow, those are hard questions. I guess I lean to the position (really a truism) that so much of mathematics as we know it is dictated by the way humans are wired – the way in which cognitive processes are rooted in vision and in physical actions, and driven by emotional needs, that it seems much safer to me to suppose these sorts of imbalances are due to human habits or characteristics. In other words, if computers were one day to do independent mathematical research, they might explore all sorts of possibilities which would seem very strange or would never occur to us, and redress some of the apparent imbalances.” —————

SH: “if computers were one day to do independent mathematical research”.
Remember that computers do what humans can do with pencil paper and eraser.
Of course they do it much faster which enables proofs which exhaust all the
possible counter-examples, proofs that might exceed a human lifetime, but
still they are in the same class of quantitative combinatorial results. So
one question is, does faster mean more intelligent? I think you mean by
“independent research”, a qualitative change in intelligence; a new range of
proofs that are not accessible or reachable by the combinatorial scaffolding
of computable proofs to which both humans and computers now eventually obtain?
I think you may be considering a non-human intelligence, not a formal system?
Anyway, I think there is a similarity between my causality quote and the
pointer you provided regarding the difficulty of how to “define” functions
and defining/creating algorithms for AI neo-theorem-proving programs. __________

http://gowers.wordpress.com/2009/06/08/why-arent-all-functions-well-defined/
“But it seems only fair, if one is going to laugh at such sentences, to provide
examples of functions that are well defined and functions that aren’t, so that
the difference can be made clear. But now we have a problem: any putative example
of a function that is not well defined is not a function at all. So it begins to
seem as though all functions are well defined. But in that case, what are people
doing when they check that a function is well defined? The real question we are
asking is this: when we use the words “well defined”, what are we referring to?”

JB mentioned the enigma of the Arrow of Time. I cobbled together this description.
‘The problem in determining the reason for the arrow of time is created by the
indistinguishability between whether time reversal non-invariant laws of nature or
temporally asymmetric boundary conditions describe reality or our perception of it.’ —————————–

“The second kind of story about how entropy increase might ground our ordinary intuitions of an arrow of time is David Lewis’s account of the asymmetry in counterfactuals.”

“The point of defending Humean Supervenience is not to support reactionary physics, but rather to resist philosophical arguments that there are more things in heaven and earth than physics has dreamt of.”

“It may be that universal history is the history of the different intonations given a handful of metaphors.” :-)

Echoes of a fractal drum,
Stephen

Restricting things to why functions are prevalent in general mathematics (rather than whether they ought to be prevalent in category theory).

I wonder if this view is partly influenced by your interest in mathematics arising from physics? I’d be inclined to say that my feeling is that the fundamental issue is not that functions are deterministic but that they are “definite decisions”. Part of my reasons is that language, from which many higher mental functions seem to be evolved, can cope with resolving quite deep “stacks” of definite elements but are very poor at resolving much less deep “stacks” which involve tracking multiple different possibilities at each level. (I know that working with relationships in mathematics would generally be dealt more by being specified by predicates, but I think this comes later than developing the comfortable “intuition” about relations.)

A more minor piece of evidence comes from the ways we model computer programs: people, particularly practicing computer programmers, are generally quite comfortable using languages/specification languages having non-deterministic choices that fit some criterion but there’s much less work (AFAIK) dealing with all the possible states consistent with the criteria. (Of course, part of that is potential combinatorial explosion, but that need not be a problem in all practical cases or when working symbolically.) One highly speculative explanation MIGHT relate this to the above language ability issue.

Bane wrote:

I wonder if this view is partly influenced by your interest in mathematics arising from physics? I’d be inclined to say that my feeling is that the fundamental issue is not that functions are deterministic but that they are “definite decisions”.

My views are very much influenced by my interest in physics. But I guess I didn’t explain myself very well: I didn’t mean that people got interested in functions because they were working on physics. What I meant is that living in a world where the future and past are so drastically different, people are bound to come up with math that has this asymmetry built in.

Part of my reasons is that language, from which many higher mental functions seem to be evolved, can cope with resolving quite deep “stacks” of definite elements but are very poor at resolving much less deep “stacks” which involve tracking multiple different possibilities at each level.

I agree, but I again see the arrow of time here. We need to predict the future much more than we need to postdict the past. Why? Because we want to keep on surviving in the future, not the past. Why? It’s an arrow of time thing.

So, trees of possibilities that branch forwards in the future are more worrisome than trees that branch backwards into the past. So, we find it very nice when a cause has just one possible effect: it simplifies our life. We don’t care as much whether an effect has just one possible cause — unless we’re doing a little piece of detective work, or history.

In short, we don’t need to think about physics explicitly to develop a preference for functions over cofunctions: the physics of the world we live in almost guarantees it.

On the contrary, it’s only when we start thinking about physics explicitly that we notice that the laws are time-symmetric, or nearly so, and we start developing an interest in categories that are self-dual. First the additive group of real numbers: the time line, which stretches symmetrically in both directions. Then the concept of groups, where every element has an inverse. Then the idea that time evolution is a symmetry, i.e. the action of a group. Then the development of matrix mechanics, where every operator has an adjoint. Then the realization that dagger-categories are important in physics, and the idea of ‘reversible computing’.

I’m exaggerating for effect here, but it would be fun to work out these ideas more carefully.

All I meant by the physics comment was that these are clearly issues at the forefront of the mind of anyone working in physics, whilst they don’t really enter the context of people who don’t work in physics who seem to display the same traits in preferring definite (sometiems arbitrary) choices. Of course earlier in the history of mathematics it was much rarer to have someone involved in mathematics who wasn’t also involved in physics in some way.

Your argument sounds very plausible, and clearly there’s a “trivial” sense in which it must be true in that it’s difficult to imagine complex physical beings arising in a universe without an initial low entropy, and human brains evolved to deal with a “deterministic at the gross level” universe. It just strikes me that from my knowledge of both geometry and computer programming, both areas where a priori there’s no direct connection with entropy, people are still very uncomfortable with dealing with many-to-many issues directly and to try to “view everything as definite choices (ie, many to one) and then fix-up multiple valuedness at the end”.

Of course, that might well be that this mental preference has evolved for the kind of reasons you say and then gets applied everywhere.

I was going to bring up the subject of cocategories at some point, and had started to jot down some references.

Right, but there’s a symmetry breaking required when we go from $\mathrm{Rel}$ to $\mathrm{Set}$, and I think that’s what David is trying to understand — that ‘force of habit’ that makes us prize functions over cofunctions.

That raises the question of why allegories are not too visible. We might have been the $n$-Allegory Café by now.

Or is it that we might as well go straight to spans, and take relation-like things to be specific kinds of span?